SMO 2023 Senior Round 1

Question 1

Find the value of $m$ such that $2x^2+3x+m$ has a minimum value of $9$.

$9/8$
$-9/8$
$81/8$
$-81/8$
$63/8$

Question 2

Which of the following is true?

$\sin105^\circ-\cos105^\circ=\sqrt3/2$
$\sin105^\circ-\cos105^\circ=\sqrt3/\sqrt2$
$\sin105^\circ+\cos105^\circ=1/2$
$\sin105^\circ+\cos105^\circ=1/\sqrt3$
None of the above.

Question 3

If $\log_{\sqrt2}x=10-3\log_{\sqrt2}10$, find $x$.

$0.32$
$0.032$
$0.0032$
$0.64$
$0.064$

Question 4

If $(x-5)^2+(y-5)^2=5^2$, find the smallest value of $(x+5)^2+(y+5)^2$.

$225-100\sqrt2$
$225+100\sqrt2$
$225\sqrt2$
$100\sqrt2$
None of the above.

Question 5

Suppose $\cos(180^\circ+x)=4/5$, where $90^\circ\lt x\lt 180^\circ$. Find $\tan(2x)$.

$24/7$
$7/24$
$-24/7$
$-7/24$
$-24/25$

Question 6

Suppose the roots of $x^2+11x+3=0$ are $p$ and $q$, and the roots of $x^2+Bx-C=0$ are $p+1$ and $q+1$. Find $C$.

Question 7

If the smallest possible value of $(A-x)(23-x)(A+x)(23+x)$ is $-(48)^2$, find the value of $A$ if $A\gt 0$.

Question 8

Find the smallest positive odd integer greater than $1$ that is a factor of $(2023)^{2023}+(2026)^{2026}+(2029)^{2029}$.

Question 9

Find the remainder of $7^{2023}+9^{2023}$ when divided by $64$.

Question 10

Let $x,y,z\gt 1$, and let $A$ be a positive number such that $\log_x A=30$, $\log_y A=50$ and $\log_{xy}(Az)=150$. Find $(\log_A z)^2$.

Find the largest integer that is less than
$$\frac{3^{10}-2^{10}}{10!}\left(\frac1{1!9!2}+\frac1{2!8!2^2}+\frac1{3!7!2^3}+\cdots+\frac1{9!1!2^9}\right)^{-1}.$$
Here, $n!=n\cdot(n-1)\cdots3\cdot2\cdot1$. For example, $5!=5\cdot4\cdot3\cdot2\cdot1=120$.

Question 12

Consider the following simultaneous equations:
$$\begin{aligned}xy^2+xyz&=91,\\xyz-y^2z&=72,\end{aligned}$$
where $x,y,z$ are positive integers. Find the maximum value of $xz$.

Question 13

Let $x$ be a real number such that
$$\frac{\sin^4x+\cos^4x}{\sin^2x+\cos^4x}=\frac8{11}.$$
Assuming $\sin^2x\gt 1/2$, find the value of $\sqrt{28}(\sin^4x-\cos^4x)$.

Question 14

A sequence $a_1,a_2,\ldots$ is defined by
$$a_1=5,\quad a_2=7,\quad a_{n+1}=\frac{a_n+1}{a_{n-1}}\quad\text{for }n\ge2.$$
Find the value of $100\times a_{2023}$.

Question 15

Let $C$ be a constant such that the equation $5\cos x+6\sin x-C=0$ have two distinct roots $a$ and $b$, where $0\lt b\lt a\lt \pi$. Find the value of $61\times\sin(a+b)$.

Question 16

In the diagram below, $CE$ is tangent to the circle at point $D$, $AD$ is a diameter of the circle, and $ABC$, $AFE$ are straight lines. It is given that $AB/AC=16/41$ and $AF/AE=49/74$. Let $\tan(\angle CAE)=m/n$, where $m,n$ are positive integers and $m/n$ is a fraction in its lowest form. Find the sum $m+n$.

Question illustration

Question 17

In the diagram below, $AB$ is a diameter of the circle with centre $O$, $MN$ is a chord of the circle that intersects $AB$ at $P$, $\angle BON$ and $\angle MOA$ are acute angles, $\angle MPA=45^\circ$, $MP=\sqrt{56}$, and $NP=12$. Find the radius of the circle.

Question illustration